Abstract. In this paper we investigate the topological properties of the set of ``fractions'' in a ``number system'' with digits $0$ and $1$ and a complex base. It is also investigated, whether the translates of this set with ``integers'' of the same number system gives a ``just touching'' covering of the plane. Algorithms are constructed to decide such questions and machine computations are also done.

AMS Subject Classification (1991): 28A80, 54-04, 11K16, 68Q20

Received June 13, 1994. (Registered under 5641/2009.)